Hubbard model¶

The Hubbard model is a simplified model used to describe the transition between conducting and insulating systems.

Version: v1 · 2026-09-28 · History
Domain-specific #: 9917
Domain group: Natural Sciences
Origin domain: Physics
Subdomains: Condensed Matter Physics, Strongly Correlated Electrons → Physics

Core Idea¶

Hubbard model is treated here as the recurring condensed-matter physics identity summarized by this source-grounded definition: The Hubbard model is a simplified model used to describe the transition between conducting and insulating systems.

The Hubbard model is a simplified model used to describe the transition between conducting and insulating systems. It is particularly useful in solid-state physics. The model is named after John Hubbard.

The Hubbard model states that each electron experiences competing forces: one pushes it to tunnel to neighboring atoms, while the other pushes it away from its neighbors. Its Hamiltonian thus has two terms: a kinetic term allowing for tunneling ("hopping") of particles between lattice sites and a potential term reflecting on-site interaction. The particles can either be fermions, as in Hubbard's original work, or bosons, in which case the model is referred to as the "Bose–Hubbard model".

For Hubbard model, the abstraction is narrower than the article's general subject matter: a positive case must preserve The Hubbard model is a simplified model used to describe the transition between conducting and insulating systems. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in condensed-matter physics, which is why this identity is domain-specific rather than prime.

Structural Signature¶

Sig role-phrases:

Defining carrier — Mathematically, the strength of this coupling is given by a "hopping integral", or "transfer integral", between nearby sites.
Constitutive relation — This coupling allows states associated with each lattice site to hybridize, and the eigenstates of such a crystalline system are Bloch's functions, with the energy levels divided into separated energy bands.
Operating condition — The width of the bands depends upon the value of the hopping integral.
Recognition evidence — The physics of the Hubbard model is determined by competition between the strength of the hopping integral, which characterizes the system's kinetic energy, and the strength of the interaction term.
Admissible variation — This orbital can be occupied by at most two electrons, one with spin up and one down (see Pauli exclusion principle).
Characteristic consequence — The first term describes the kinetic energy of the system, parameterized by the hopping integral, t.
Failure boundary — The one-dimensional Hubbard model was solved by Lieb and Wu using the Bethe ansatz.

What It Is Not¶

Not the whole field of condensed-matter physics. The node requires the specific identity stated by The Hubbard model is a simplified model used to describe the transition between conducting and insulating systems.
Not an over-broad reading. If U/t is not too large, the overlap integral provides for superexchange interactions between neighboring magnetic moments, which may lead to a variety of interesting magnetic correlations, such as ferromagnetic, antiferromagnetic, etc. depending on the model parameters.
Not an over-broad reading. Although Hubbard is useful in describing systems such as a 1D chain of hydrogen atoms, more complex systems may experience other effects that the Hubbard model does not consider.
Not an over-broad reading. However, it is possible for the electrons to exhibit another kind of behavior.
Not automatically Particle in a one-dimensional lattice. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Hubbard model applies literally inside condensed-matter physics wherever the source-defined carrier and relation can be established. Its documented habitats include:

Narrow energy band theory. This coupling allows states associated with each lattice site to hybridize, and the eigenstates of such a crystalline system are Bloch's functions, with the energy levels divided into separated energy bands.
Numerical treatment. DMFT allows one to compute the local Green's function of the Hubbard model for a given U and a given temperature.
Simulator. A "backwards" stacking regime allows the creation of a Chern insulator via the anomalous quantum Hall effect (with the edges of the device acting as a conductor while the interior acted as an insulator.) The device functioned at a temperature of 5 Kelvins, far above the temperature at which the effect had first been observed.
Narrow energy band theory. When the Hubbard model is used to describe electron systems, these interactions are expected to be repulsive, stemming from the screened Coulomb interaction.
Narrow energy band theory. For example, it has been used to describe metal oxides as they are heated, where the corresponding increase in nearest-neighbor spacing reduces the hopping integral to the point where the on-site potential is dominant.
Narrow energy band theory. The density operator is \hat{n}i = \hat{n}_i \vert \Phi \rangle.} + \hat{n}_{i\downarrow} and occupation of i -th site for the wavefunction \Phi is n_i = \langle \Phi \vert \hat{n

Outside condensed-matter physics, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Theory or should be marked as analogy.

Clarity¶

A clear use of Hubbard model names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is The Hubbard model is a simplified model used to describe the transition between conducting and insulating systems. The strongest recognition evidence in the frozen account is: The physics of the Hubbard model is determined by competition between the strength of the hopping integral, which characterizes the system's kinetic energy, and the strength of the interaction term. A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification If U/t is not too large, the overlap integral provides for superexchange interactions between neighboring magnetic moments, which may lead to a variety of interesting magnetic correlations, such as ferromagnetic, antiferromagnetic, etc. depending on the model parameters. so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

Hubbard model compresses multiple condensed-matter physics details into a stable diagnostic relation. The source shows both the central mechanism—this coupling allows states associated with each lattice site to hybridize, and the eigenstates of such a crystalline system are Bloch's functions, with the energy levels divided into separated energy bands.—and the practical consequence—the first term describes the kinetic energy of the system, parameterized by the hopping integral, t. This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning¶

Type the carrier. Identify the condensed-matter physics entities to which the claim applies.
State the relation. Use the source-grounded identity: The Hubbard model is a simplified model used to describe the transition between conducting and insulating systems.
Check operation and conditions. The width of the bands depends upon the value of the hopping integral.
Demand recognition evidence. The physics of the Hubbard model is determined by competition between the strength of the hopping integral, which characterizes the system's kinetic energy, and the strength of the interaction term.
Test variation. Change an implementation or setting while preserving this orbital can be occupied by at most two electrons, one with spin up and one down (see Pauli exclusion principle).
Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Theory.

Knowledge Transfer¶

Within the home domain. Knowledge about Hubbard model transfers literally when a new case preserves the same carrier type, relation, and recognition test. This coupling allows states associated with each lattice site to hybridize, and the eigenstates of such a crystalline system are Bloch's functions, with the energy levels divided into separated energy bands. DMFT allows one to compute the local Green's function of the Hubbard model for a given U and a given temperature.

Beyond the home domain. Transfer the broader Theory relation when the condensed-matter physics-specific differentia cannot be filled. Retain the name Hubbard model only when the same carrier, operation, and rejection conditions are present literally rather than metaphorically.

Examples¶

Canonical¶

For example, the Hubbard model correctly predicts the existence of Mott insulators: materials that are insulating due to the strong repulsion between electrons, even though they satisfy the usual criteria for conductors, such as having an odd number of electrons per unit cell. This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → The Hubbard model is a simplified model used to describe the transition between conducting and insulating systems; recognition evidence → The physics of the Hubbard model is determined by competition between the strength of the hopping integral, which characterizes the system's kinetic energy, and the strength of the interaction term

Applied / In Practice¶

For example, it has been used to describe metal oxides as they are heated, where the corresponding increase in nearest-neighbor spacing reduces the hopping integral to the point where the on-site potential is dominant. The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → Narrow energy band theory; invariant → The Hubbard model is a simplified model used to describe the transition between conducting and insulating systems; boundary → the case exits the class when if U/t is not too large, the overlap integral provides for superexchange interactions between neighboring magnetic moments, which may lead to a variety of interesting magnetic correlations, such as ferromagnetic, antiferromagnetic, etc. depending on the model parameters

Structural Tensions¶

T1 — Stable identity versus admissible variation. If U/t is not too large, the overlap integral provides for superexchange interactions between neighboring magnetic moments, which may lead to a variety of interesting magnetic correlations, such as ferromagnetic, antiferromagnetic, etc. depending on the model parameters. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. Although Hubbard is useful in describing systems such as a 1D chain of hydrogen atoms, more complex systems may experience other effects that the Hubbard model does not consider. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. However, it is possible for the electrons to exhibit another kind of behavior. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. Unlike Mott–Hubbard insulators electron transfer happens only within a unit cell. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. Mathematically, the strength of this coupling is given by a "hopping integral", or "transfer integral", between nearby sites. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate Hubbard model literally, co-instantiate Theory, or only resemble it?

T6 — Autonomy versus reduction. This coupling allows states associated with each lattice site to hybridize, and the eigenstates of such a crystalline system are Bloch's functions, with the energy levels divided into separated energy bands. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Hubbard model distinguish that the broader parent Theory leaves together?

Structural–Framed Character¶

Hubbard model is structural-leaning. Its structural side is the repeatable organization summarized by The Hubbard model is a simplified model used to describe the transition between conducting and insulating systems. Its framed side is the condensed-matter physics vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: The width of the bands depends upon the value of the hopping integral. Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Theory. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent¶

What is skeletal. The Hubbard model is a simplified model used to describe the transition between conducting and insulating systems. The reviewed portable genus is Theory; the candidate preserves that parent relation across admissible variants. The source-grounded carrier and relation are expressed by these conditions: Mathematically, the strength of this coupling is given by a "hopping integral", or "transfer integral", between nearby sites. This coupling allows states associated with each lattice site to hybridize, and the eigenstates of such a crystalline system are Bloch's functions, with the energy levels divided into separated energy bands. The recognition and variation tests add: The width of the bands depends upon the value of the hopping integral. The physics of the Hubbard model is determined by competition between the strength of the hopping integral, which characterizes the system's kinetic energy, and the strength of the interaction term.

What is domain-bound. condensed-matter physics fixes the carrier, technical vocabulary, admissible evidence, and exceptions that distinguish Hubbard model from other Theory instances. Its documented habitat includes the condition that This coupling allows states associated with each lattice site to hybridize, and the eigenstates of such a crystalline system are Bloch's functions, with the energy levels divided into separated energy bands. A second source-grounded application condition is that DMFT allows one to compute the local Green's function of the Hubbard model for a given U and a given temperature. Those details determine what the words denote, what observations warrant classification, and which apparent similarities are false positives.

Why the node remains domain-specific. Removing the condensed-matter physics differentia leaves the parent rather than the candidate. The edge records that reduction without claiming that every topical neighbor is hierarchical. The final collapse test is source-specific: This orbital can be occupied by at most two electrons, one with spin up and one down (see Pauli exclusion principle). If that condition or the defining relation is absent, the case may instantiate Theory, but it is not Hubbard model.

This entry is a kind of Theory.

Immediate parent — Theory (subsumption). Hubbard model is a domain-specific kind of Theory. Hubbard model is a strict kind of Theory: The Hubbard model is a simplified model used to describe the transition between conducting and insulating systems. The parent supplies the necessary broader identity—A coherent system of concepts and propositions that explains, organizes or predicts a domain through explicit relations and standards of support.—while the candidate adds its domain carrier, relation, and rejection conditions.
Other nearby abstractions. Retrieval neighbors remain comparison surfaces only; no additional parent is asserted without a necessary-genus or structural-prerequisite test.

Relationships to Other Abstractions¶

Current abstraction Hubbard model Domain-specific

Parents (1) — more general patterns this builds on

Hubbard model is a kind of Theory Prime

Hubbard model is a strict kind of Theory: The Hubbard model is a simplified model used to describe the transition between conducting and insulating systems.

Hierarchy paths (2) — routes to 2 parentless roots

Hubbard model → Theory → Formalization → Representation → Abstraction

Show alternative path (1)

Neighborhood in Abstraction Space¶

Hubbard model sits in a moderately populated region (52^nd percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Condensed Matter & Physical Chemistry Models (26 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Theory. The parent omits the specialist differentia. Tell: Can the case establish The Hubbard model is a simplified model used to describe the transition between conducting and insulating systems?
Particle in a one-dimensional lattice. The quantum model of a particle moving in a spatially periodic one-dimensional potential, whose stationary states have Bloch form and organize into energy bands separated by gaps. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Bohr model. A historical atomic model with electrons restricted to discrete stationary orbits and emitting or absorbing photons only when transitioning between quantized energy levels. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Ising model. A statistical-mechanical model of binary spins on a graph whose energy rewards or penalizes neighboring alignment and external-field orientation, exhibiting collective order and phase transitions. Tell: Which entry's carrier, operation, and failure condition are satisfied?
A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Hubbard model remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside condensed-matter physics lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Theory?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Hubbard_model (revision 1368000717).
Preserved source candidate: https://books.google.com/books?id=0KMkfAMe3JkC&pg=RA4-PA58
Preserved source candidate: https://royalsocietypublishing.org/doi/10.1098/rspa.1963.0204
Preserved source candidate: https://www.quantamagazine.org/physics-duo-finds-magic-in-two-dimensions-20220816/
Preserved source candidate: https://archive.org/details/arxiv-cond-mat0610710
Preserved source candidate: https://archive.org/details/arxiv-1505.02290
Preserved source candidate: https://books.google.com/books?id=uCPgHgEKnwEC&pg=PA1

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.